What are the six trig function values of #5pi#?

1 Answer
sente
Dec 31, 2015

#sin(5pi) = 0#
#cos(5pi) = -1#
#tan(5pi) = 0#
#sec(5pi) = -1#
#csc(5pi)# is #"undefined"#
#cot(5pi)# is #"undefined"#

Explanation:

First, let's note that sine and cosine are periodic with period #2pi#, meaning for any integer #k#,
#sin(x) = sin(x+k2pi)# and #cos(x) = cos(x+k2pi)#
With this, we get the first two desired values:

#sin(5pi) = sin(pi + 2(2pi)) = sin(pi) = 0#
and
#cos(5pi) = cos(pi+2(2pi)) = cos(pi) = -1#

The remaining trig functions are just functions of sine and cosine.

#tan(5pi) = sin(5pi)/cos(5pi) = 0/(-1) = 0#

#sec(5pi) = 1/cos(5pi) = 1/(-1) = -1#

#csc(5pi) = 1/sin(5pi) = 1/0# is #"undefined"#

#cot(5pi) = cos(5pi)/sin(5pi) = (-1)/0# is #"undefined"#

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